ERRATA IN DO CARMO, DIFFERENTIAL GEOMETRY OF CURVES ...

• p. 155: It may be preferable to write equation (3) in the form( ) ( ) ( )e f E F a11 a− =12,f g F G a 21 a 22(the transpose of the current version of the equation), so that the matrix (a ij ) thatappears in it is exactly the matrix of dN p , instead of its transpose.• p. 155, middle, just before the formula for a 11 : Each x in {x u , x v } should be bold.• p. 161, line 2 of Example 4: change “changed” to “replaced”.• p. 162, middle, “If a parametrization of a regular surface is such that F = f = 0, thenthe principal curvatures are given by e/E and g/G.” A more enlightening explanationof this is given by the equation following equation (3) on page 155.• p. 214, first display: Some indices are backwards (the ij-th entry of a matrix istraditionally the entry in the i-th row and j-th column, and in fact this conventionis followed on page 154). Change this line to the following:α ij = 〈Ae j , e i 〉 = 〈e j , Ae i 〉 = 〈Ae i , e j 〉 = α ji ;• p. 215, Lemma: The letter “a” of the word “at” should be italicized.• p. 221, top right of Figure 4-2: dφ p (W ) should be dφ p (w).• pp. 224–225: The function F is being confused with F ◦ x (which appears on page225). For F to be a function from U to R 3 , it should be defined as a function of xand y. For F ◦ x to make sense, F should be a function of x, y, z. But F was definedas a function of ρ and θ!• p. 226, display in Definition 3: Since the subscript p is used on the inner product onthe right, it would make sense to use the subscript ϕ(p) on the left.• p. 226, second to last display: The range for θ should be 0 ≤ θ ≤ π.• p. 227, sentence after Proposition 2: With the given definition of local conformality,it seems to be very difficult to prove that it is a symmetric relation, unless oneaccepts the unproven theorem on page 227. Also, the sentence as written misleadinglysuggests that transitivity is the only property required for an equivalence relation.• p. 232, end first paragraph: Italicize the first appearance of “Christoffel symbols”instead of the second.• p. 234, sentence after Theorema Egregium: Presumably the codomain of ϕ was supposedto be a possibly different surface ¯S. In this case, the S near the end of thissentence also should be ¯S.• p. 238, sentence before Definition 1: Start this sentence with “The vector field” (toavoid starting it with the mathematical symbol w), and change “for every p ∈ U” to“at every p ∈ U”.• p. 239, middle: “Sec. 4-1” should be “Sec. 4-3”.• p. 239: In (1), the functions a and b depend on the curve α, so it should be explainedthat the values of a ′ and b ′ depend on α only through its tangent vector.• p. 243, Figure 4-12: φ should be ϕ.• p. 245, Figure 4-14: φ should be ϕ (4 times).• p. 245, last line of Definition 8: “for all t ∈ I” should be “at all t ∈ I” (to match theprevious part of Definition 8).• p. 246, Example 3: “there exists exactly one geodesic C ⊂ S passing through p andtangent to this direction”. Strictly speaking, any connected open neighborhood of4